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EXAMPLES 



IN THE 



INTEGRAL CALCULUS, 



COMPILED FOR 



THE USE OF THE CADETS 



AT THE 



UNITED STATES NAVAL ACADEMY. 



■ » 






WASHINGTON: 

GOVERNMENT PRINTING OFFICE 

1874. 









A 



\> 



> 



THE INTEGRAL CALCULUS 



DEFINITIONS. 

1. In the differential calculus, we have to find the rates, or 

differentials, of given functions. 

In the integral calculus, having given any differential, we 
have to determine the function of which this is the rate ; this 
function is called the integral of the given differential. 

The integral sign is / 5 it is always written before an 

expression for a rate, the rate being generally expressed by 
a single variable and its differential; as, 

x 2 dx 

which means that function of a? whose rate is x 2 dx. 

2. A definite integral is written with a number or letter, 
denoting a special value of the independent variable, at the 
bottom, and another at the top of the integral sign (these 
letters or numbers being called limits) ; thus, 

»3 

x 2 dx 
i 

and indicates the amount by which a quantity, varying with 
the rate under the integral sign, actually varies while the inde- 
pendent variable passes from the lower to the upper limit- 
Thus, the above expression denotes the increment received by 
any quantity which has the rate x 2 dx while x increases from 
1 to 3. 

3. An indefinite integral is simply a variable which varies 
with the rate expressed under the integral sign, and is writ- 
ten without limits ; thus, 

x 2 dx 



I; 



/• 



It will be seen that the definite integral may, then, be 
defined as the increment received by the indefinite integral 
while x passes from the lower to the upper limit. The in- 
definite integral is thus a function of co, while the definite 
integral, being the increment received while x passes from 



one limit to the other, is a function of these limits or special 
values of x. The indefinite integral, being then such a func- 
tion as will have the given rate, will be known if we can 
recognize the differential as one derivable from any particu- 
lar function. 

Thus, d (^x 3 ) = x 2 dx; hence ^ x 3 is an iutegral of x 2 dx, i. e., 
a quantity which has that rate; and 



f 



x 2 dx = 8| 



i 

because J a? 3 increases from £ to 9 as x increases from 1 to 3- 
The indefinite iutegral is indefinite, not only because it is a 
variable or function of a?, but also for the following reason: 
^ x 3 is not the only quantity which has the rate .r 2 dx; for if 
c denote a constant, J* 3 + c has the same rate; hence the 
indefinite integral 



/ 



= ^ x 3 -f c 



which includes all functions of x which have the rate x 2 dx. 
c is wholly iudefinite in value, and is called the constant of 
integration. 

-4. The definite iutegral is derived from the indefinite by 
computing the value for the upper limit and subtracting from 
it that for the lower limit: this, evidently, is the increment 
received by the indefinite integral as x passes from the lower 
to the upper limit, 

»3 

dx = l-.:p-%.V = 



: 



p 



3 

It will be seen that, if we had used the general form 

.r 2 dx = ^ as 3 + c 

we should have obtained the same value, c disappearing by 
subtraction. 

o. Aparticular integral is a particular value of the indefinite 
integral obtained by giving to the constaut e some special 
value. The particular value of the constant is usually deter- 
mined by the condition that the integral shall have a certain 
value for a given value of the independent variable. Thus, 
the space through which a body falls freely front rest in t 
seconds is a particular integral of gtdt, this being the expres- 
sion for the rate at any time t (from rest). It is. in fact, that 
particular integral which is zero when t = 0, since the posi- 



tioD of rest is our origin of distances. Tbis particular inte- 
gral is denoted by 

f Q 9tdt 
therefore we write 

8 = I gtdt 

to express that s is measured from the position of rest. An 
integral with a lower limit expressed, but without an upper 
limit, is then a function of the independent variable, and is 
the same as the indefinite integral minus its value at the 
lower limit, the constant of integration being thus eliminated. 
Thus, 

r t dt = J t 2 + c 



1- 



tdt = l> t 2 

'0 

hence, in above example, 

s = igt 2 
©. From the notation employed for definite integrals, we 
easily see that 

Xa nb 

f{x) dx = — I f(x) dx 

J™b fc re 

f(x) dx 4- I f(x) dx = J f(x) dx 
a *Jb U a 

f a f(x)dx = 



and 



ELEMENT AKY PROPOSITIONS liELATiVE TO INTE- 
GRALS. 

7. The three elementary propositions of differential calcu- 
lus; viz, 

I, d (x -f h) = dx 

II, d {mx) = m dx 

III, d (x + y + &c.) = dx + dy + &c. 
give three propositions respecting integrals. 
I shows that 



f dx= f' 



d (x -f h) = x 4- h 

dx = x + c 



/• 



6 

that is, points out the necessity of adding a constant of inte- 
gration to get the general value of the indefinite integral. 



II gives 



b 

dx 



Jm dx = mb — ma = m (b — a) = m J 

from which, as x may represent any variable, we find that a 
constant factor under the integral sign may be removed and 
written as a coefficient of the integral ; thus, 



I m dx = m I dx 



We may also, when desirable, introduce a constant factor 
under tbe integral sign and its reciprocal outside of the inte- 
gral sign, for 

/dx = I — m dx = — / m dx 
J m m J 

thus 

/ x* dx = i / 5j? 4 dx = } .r 5 + c 

since we know that 

d (xf = ox* dx 
III gives the following principle of integration, since 

d (x + if + &c.) == dx + dy + &c. 

/ {dx + dy 4- &c.) = / dx 4- J dy + &c. + c 

A constant of integration may be considered as arising 
from each term of the second member, all of which constants 
unite into a siugie one. This expresses that a polynomial 
differential expression is integrated by treating each term 
separately. In the above expression of the principle, it is 
impossible to use limits, as the same limits cannot be values 
of each of the variables; but if, which is generally the case, 
we have a series of terms, each a function of the same 
variable, limits may be used on both sides ; thus, 

Jr>b 
(log x -f m cos x 4- 5) dx = 
a 

Jfb Pb 

log x dx 4- m j cos x dx -f 5 (b — a) 

FUNDAMENTAL INTEGRALS. 

8. The fundamental differentials give the following inte- 
grals, on which all other integrable forms depend. The con- 



stant of integration, though omitted hereafter, must be con- 
sidered as implied in all indefinite integrals. 

1. I X'" dx 



m + 1 
o. I a* ax 



logwt 



8. j tan x sec a? <7a? = sec x 

9. / cot x cosec x dx = — cosec a? 



4. I cos a? da; = sin x 

5. / sin x dx = — cos x 

6. / sec 2 a? da? = tan a? 

7. j cosec 2 a? tfa? = — cot 
/ tan a? s 

/- da? 1 . pwaq 

10. / — : — - — ■ — z-r = — sin -1 — 
/ v(ft — war) m L a J 

u-r-r? L ==-t»^r-i 

J cr -|- nrar am |_ a J 

12 r <fa 1 gec _ 1 pw£~| 

"" j xV{ni 2 x 2 — a 2 ) ~~ a [_ a J 

„ _, p dx 1 . Y~mx "I 

13. / -77^ 2\ = -r vers — 

J \/(2ax — mar) v«* L a J 

Equation (1) is to be used when the value of m is fractional 

or negative, except in the case where m = — 1, when it fails 

to give an intelligible result, and equation (2) must be used. 

DIRECT INTEGRATION. 

0. An expression is said to be directly integrable when it 
obviously corresponds in form with one of the above funda- 
mental integrals, some other variable and its differential 
merely taking the place of a? and dx; for example, 

/'cos x dx rd(smx) _ . ■ 

— ; = I — ^ = log (sin x) 
sm x J sin x 

It is also said to be directly integrable when it can be made 



8 

to correspond to one of tlie fundamental forms by merely 
introducing a constant factor under the integral sign ; thus, 



l y/(a — bx 2 ) x dx = — ~r I (a — &as?)*(— 2bx 



dx) 



1 a 

= — tw (a — bx 2 )* 
3b 



EXAMPLES. 
I. 



v,3\3 



1. I xV (a 2 + x 2 ) dx = £ (a 2 + x 2 Y 

_ C x dx , i 

2. I = log (a 2 + x 2 ) - 

J « 2 + x 2 * v y 

_, C <*'~ ^ r , 1 
o. j r = log 

5. fx V{a* - b*x-) dx = -^ (<i 2 - 6V) 
J .f- x 

7 . r *■ _ * 

8- f , ^ , •= Vfft' + g 2 ) 

10. f(a + 6a?) 2 dx = ^ [a + bx) 2 

11. f(« + 3,r 2 ) 3 xtf.r = — (a + 3r 2 ) 4 

12 f- = - — 

13. f(a + kr» + l )»-> *■ to = (^t^l): 1 
J n(n+l)b 

/Off y Tf>2 

■ f(--~ 

J \ X 2 X 



15. \^*\U=\-^ 



fi y>3 



X° X 



9 



10. 
17. 
IS. 
19. 

20. 
21. 



2j(x)dx , 1 



J 1-xl 



7x dx . 1 

■= log 7^ 



Sa — ox 2 ~ ~° (8« — 30?)% 
(2a?* + j?- 1 ) dx =—— 4 log* x 

f f ~ 4 f + f cfc = log (x 3 - 6a; 2 + 9x) { 
J x 3 — Oa? 2 4- 9a? v 



/ ( « — — 4- ex' 1 } dx = 



2ca? 



x 
x n ~ l dx 



= aw+ —=- + 



2x 2 



/x h 
{a 4 bx n )'"~ n b (m — 1) (a + bx n ) m ~ 1 
' (a — x) dx 



rja- 
J VP 



- = V (2ax — X' 2 ) 
«a? — x £ ) 



24, 



L'J, 



20, 



/* 5 9 

I 3 (a 4 — a' 4 ) 3 a? 3 (?a? = — — (a 4 — x*) 

fa 2 + 2ax , ^ . 
— : r- «j? = 2a v (aa? 4- ar ) 
V(aj? 4 ^ ,2 ) 

J sin (2a?) (7a? = — J cos (2a?) 
J sin 3 j? cos x dx = 
T(l + 3s 



sin 4 a? 



sin 2 a?) 2 sin x cos a? (far = 



2 ^.\3 



(14-3 sin 2 a?) 
18 



28. 



sin xclx 



COS a? 



log (sec x) 



f< 



(a 2 4- ?> 2 sin 2 a?)* sin a? cos x dx = 



(a 2 4- J» 2 sin 2 a?)* 



36 s 



34. 
35. 



/. . . „ , . , cos 4 (wa?) 
sin (wjp) cos 3 (wa?) (7a? = — - 
v ^ v ; 4w 

C • n / v / x ^ siu w+1 (-/ia?) 

| sin' 1 (?iar) cos (wa?) (/a? = — ' 

J K v ' n(n+l) 

31115 (I) co8 (§)* r =* 8in6 (i) 

/a? (7a? 
— t— - = J tan (a? 2 ) 
cos 2 (a? 2 ) v 

tan 2 a? (7a? 



30. 



31. 



32. j si 



cos^a? 



= J tan 3 a? 



| sec 4 a? sin a? dx = ^ sec 3 a" 



36. f< 



10 

cosec 3 x cos x dx = — J cosec 2 a? 
fsec 2 .z . 

38. i vers # sin x dx — ^ vers 2 x 

39. 1(1 — cos 2 x) sin a? tfa? = — — — — cos x 

/siii^ 
(sin 3 a- — cos 3 x) sill a? cos .r dx = ! 

41. I ,— = ^-~- Ti = — 4 (a* — o sin x)* 

J (a 2 — 3 sin x)h 3 v J 

42. / 6 sec (±x) tan (4#) efa? = | sec (4#J 

43. / 2 sid (a + 3x) dx = — § cos (a + 3«r) 

M .j3io^ = (Ioga;)3 

© 

= — 2 cosec ( - J 




= 4 sec* a- 

COS° # 



/cos x dx , 1 
; = l0g ; 
a — sin x a — si 



sin x 
sin x dx 1 



(a — cos x) 2 a — cos x 

n rsin 2 x cos x dx 2 xi 

J (a 2 — b sin 3 #) 3 3ft v 

a 3 * <ta = — : 

3 log a 

51. Ce^xdx = Je 2 * 2 

52. f e(* 3 -* 2 ) (3a? - 2a?) tte = e (* 3 -* 2 > 

X 

53. I a a dx=- 

J log a 

/m 
m e-™ dx — e~ nx 
n 



11 

55. / (e x — e~ x ) dx = e x + e~ x 



56. f ~ dx = log (e* + e~ x ) 

57. fe sm x cos x dx = e sin K 

58. / e- cos2:c sin a? cos a? t7a? = J e~ cos2 
. | (e 2 * — e-tx) dx = J (e 2 * + e-*») 

s 

— 
a 



59 

60. / e seca: sec 2 x sin x dx = e sec x 



X 

V— tan -1 - //t p—i&xv— 1 - 

r -f- # ft 

62. f (1 - a 1 ) 2 a* Ab = - (1 , ~ a ^ 
J 3 log a 

63. 



3. f (3a; 2 - e*) f (6* - e) dx = 2 ( Sa * J~ ^' 
64. I 

ft cos (3.) gjjj ^3^) ^ _ _ 



•cos (3x) 



log a 



a(«+&«) 
66. I ««*+&*> <fa? 



/t. 
6 log a 

/* dx 1 . _ y /bx\ 

J (ft 2 — b 2 x 2 )* ~ b "" \ ft / 

68. / - ~— L = sur 1 ( — - — ) 

J \ a -{x + bf\i V Vft y 

69 - r, ^ « = -75 tan- 1 (a? V5) 

J 1 + 5f V5 v 7 

_ f 2r7* 2 , ( /3) 

J a? (3ar — 5) 2 y^> / v5< 

_, f 2a?- 1 da? 2 , ( /14, 

n -J- { ux* - 3)* = v^ cosec j V t! 

_ rt /' x 2 dx H , _ 

72. / — — = i- tan- 1 a; 3 
J a? 6 -f- 1 3 

73. r_^_ = j 8 i n -i iC 3 



12 



„ r xdx i . _ a /&a? 2 \ 

„ P dx .(2x) 

75. I ^r = vers -1 < — > 

J {ax — x 2 f ) a V 

76. f *— 

J (4ax — x 2 ) 

r dx 1 

"'J (2flff - b 2 x 2 f- = I 

78. f 3(U 3 

79./ 



_ vers -i ) j 



T vers - 
b 



. \SaXi 

— -"~ • vers — < 

(?> 2 a? — 4«a" 2 )* 2 \/a ' ) 6 2 < 



(&2 _ 3.r 2 )* i/3 



1 . A^V^ 
1 = -— sin-W-^ 



82 . (" **■ * ^(fT) 

J ar (ar — a 2 ) 6a \a J 

fx dx 1 _ T / 6a: 2 \ 



83 

84 

85 
86 



/ affix 1 A Z>a? 3 

= log sin x 

tana? & 

■/ 



e* -f e~ 
dx 



= tan -1 e x 



87 -JVi^ =,og(,oga!) 

/cosa? tfa? . , (sin 

7— 9 — 9 — tt = sin -1 { — 
(a 2 — sura?) 2 f a 

h 



88 
89 



cos x dx n X( 

dx 11 /18a?\ 

aa? — 9a? 2 )^ " \ 7a J 



f Sadx _3a _, /3a?\ 
' °* J a; (9a. 2 - 4)* "" 2 SeC ^2 ) 
C sin a? tfa? 1 

' J (1 + cos a?) 6 ~ 5 (1 + cos a?) 5 



92. J (cos x + sin xf (cos x — sin x) dx = 
P dx 1 . , { .. i^ 



IS 

(cos a? + sin ;z) 4 



94. I -7 — = - T sin-W.» , 

— — — = i 102: tan x 
sin 2# 

06. f-= - 

J .) (n — l).r-' 

n ^ f\ dx dx / , a 4- x 

m + x b -\- ,r\ b + x 



98. I sin 2a? r7,r = sin 2 x 1 



rv.r I X 

dx 1_ 

(2a? — 5.^ ~ "75 



100. / - (lv ■= -L vers- 1 j 5a? j 

J 2# — 5,r 2 H V-> 



INTEGRATION BETWEEN LIMITS. 

EXAMPLES. 
II. 



f V ^7,r = 



1. I j? 5 rf.r = 10.5 



2. f (,:r — ,i 2 ) r7a? = f 
J 



J cos 

Jo 
no 

4. Ia{a + bxfdx= — 



cos « rf.r = 1 



, >a dx 



«-f. 



V {a 2 - x 2 ) 2 

a dX TV 

a 2 + a? 3 ~~ 4& 



14 



_ r 4 sin x dx 1 . 

— = - log 2 



COS X 



dx 



8. / - 
Jo (ff — a - a 

J rt J. 1 — 

10 . r * « 

J fl 30 + 4r 48 

11. / —— - £. = — a 

Ji / 3 

13. f \x* - x* + x>\flx = 

J, 16 + 9** 24 



29 



105 



= 1 



«_ — hi 



16 



r** xdx 1 

'jo cos : r ~J 



J 14 7 

cos- x sin .r <7r = — 
o 48 

J^ 1 

sin 4 x cos .r ox = - 
o 5 

19. f sini2.n cos _ dx=-= 
Jo 4 

20- / T^ i = r> og 4 

Jo 3^~ + 

J^ l 
e-' = 

o 

ir 

22. f V* P"« cos (2sr) <?.r = ^^ 
Jo (a 3 — a..r)^ 6V'« 



15 



INTEGRATION BY CHANGE OF FORM. 

lO. A quantity may become directly integrable by a mere 
change of form ; thus, 
By expansion, 

1(1— x 2 ) 2, dx — I dx — 2 / x 2 dx -f I x*dx 

the terms of which are directly integrable. 
By division, as 

— I — x 2 dx — I ax dx — a 2 I dx — a 3 I 

a — x J J J J x — a 

By separating the parts of a fraction, as 

J^a + ox , _ C dx r* x dx 

a 2 -\- x 2 J a 2 + x 2 J a 2 -j- x 2 

By adding and subtracting the same quantity; thus, 

/ y/(a — x) x dx = — i -\/(a — x) \(a — x) — a\ dx 

= — j (a — xY dx + a I V(«, — x) dx 

This is equivalent to separating into parts, which have been 
disguised, by the cancellation of equal positive and negative 
terms. 

By multiplying numerator and denominator by the same quan- 
tity ; thus, 

'^(x 2 — a 2 ) p x? — a 2 



Xy/{X 2 — a 2 ) 

x dx „ /* dx 



J V(x 2 — a 2 ) J x^/(x 2 — a 2 ) 
By trigonometrical transformation, as 

I sin 3 x dx = I sin x (1 — cos 2 x) dx 

= / sin x — J cos 2 a? sin a? <fa? 



EXAMPLES. 



TIL 

= — a 5 

. 4a,- 5 + 5x 3 . t — + 20a; 4 + 535a- 3 + 16050a; 2 
2. I — — dx = j 5 



. f a (3x + 2a) 2 x 2 dx 

5. P* 5 ± 
t/ a? — • 



20 £ + 042000a; -f 12840000 log (x - 20) 



16 



3. f%? + 3a) 3 x 5 dw= 9fff a £ 

'J 



3x 3 + 4a? 2 + 2a? + 3 



x 2 + 1 



= — -f 4a? — J log (a? 2 + 1) — tan -1 .*? 



•I. 



4- a? 



a 2 — x 2 ) 



— dx = a sin - 



a;' 2 rfa? 



<z 2 + x 2 

(a + * 

(a — x) 



= x — a tail' 



u -i (-) - x /(a 2 -x 2 ) 

-'©" 



7. / 1— "— \ dx = / T ~ x dx 



= a sin x ( - ) — V(« 2 — a? 2 ) 



J a? 



2aa? — a? 



- dx 



x\/('2ax — a?'-') 

J* (a — x) dx P a dx 

^/{2ax — x 2 ) J y/{2ax — x 1 

— s/(2ax — x 2 ) -f a vers -1 ( - J 
10. / \/(a? + a) x dx = I y/(x + a-) j(a? + a) — « j 



'7a? 



3 1 x 

(a? -j- &)* dx — la (x -f- tt ) ^ 



2(a? + a)" 2«. (a? + a)^ 



3 



/«J? i '|^' ' . •> ~Y' — — — •■'~ii 

(a-bx-x*f = J \_ a + J - (y + x ; J 

• _! h + 2a? 
~ felU V(4a + 6"2) 

I c< 

Jo 

3. J, 



12. 1 cos 5 a? da? = T 8 3 



13. I sm 3 OGos?0dd 



14. |sec 4 dd = tan -f 



cos 6 cos 4 fl 
tanty 



17 



/tan 2 tan 4 
sec 4 tan d0 = — 1- — - — 
2 4 

16. |sec 6 d0 = f (tan 2 e + I) 2 sec 2 do 

2 tan 3 tan 5 
= tan + 



3 



r & f— ^ 

17. I - — = I o • ^ a? 
J sin a? I 2 sin - cos _ 



, X (AX 

sec — . — 

2 2 




20. | tan 3 x doc = — - — + log cos a? 



sin z x cos^ a? 

= tan x — cot a? 

* = /' V(» + a) - V(« + &) & 



V(# + «) + -/(# + b) J a — b 

9 (x + a)* - {x + &)* 



~ 8 a-6 

$a? 

23. I z : — = tan x — sec x 

+ sin x 



•h 



dx 
1 + cos x 

dx 1 [*{a + for 3 ) — for 3 



24. I - — — - = cosec a? — cot x 



25 . r «^_ = if i 

J a? (& + for 3 ) aj 



x (a + for 3 ) 



(7a? 



1 \ ar 

x- l0£ 



3a ° ( a-\- bx 2 

nn r dx b _ \a + bx 3 

26- J ^TT-K^ = £5 l0 S 



a? 4 (a + 6a? 3 ) 3d 2 3 > a 3 V 3a x : 
3 



18 

o- f (lx _ ] i „{ •** ? 

— — = — ! : = logtan} T + £$. 

cos.r in ) M 2\ 

Janj--»j ( 

30. r f7j r (lr 

J sin x -f- cos x J sin x + sin (90 — x) 



dx 



1 _ U -, 

= -7- loo- tan <'_ + -' 



INTEGEATIO^ OF EATIOXAL FBACTIOXS. 

11. Any fraction whose terms are rational functions of x 
can be integrated, provided the denominator can be resolved 

into factors of the first and second degrees. 

If the numerator is of a degree equal to or higher than that 
of the denominator, the fraction must be reduced by division 
to the form of a mixed quantity. The entire part will be 
directly integrable. We, therefore, have to consider those frac- 
tions only whose denominators are of a higher degree than 
their numerators. The denominator is supposed to be resolved 
into factors of the first degree, and quadratic factors which 
cannot be resolved into real factors of the first degree. The 
latter can be put in the form 

(ax + h) 2 4- c#2 
The following examples will serve to explain the method 
of decomposing into rational fractions, and of integrating the 
results. 

EXA3IPLES. 



IT. 
x — 1 
x 2 + 6x + 8 
To decompose, put 

x-1 A B 



!• J 7^~- 



+ 



(x + 2) (x + 4) x + 2 ' x + 4 



19 

whence 

x - 1 = A (x + 4) + B (a + 2) 

which must be true for all values of x. Then, to determiue 
A, let x — — 2, which will make the term containing B dis- 
appear • thus, 

- 3 = A (2) 

or 

j 3 

To determine jB, let a? = — 4 : 

- 5 = #(-2) 
or 

Then 

a? — i , „ r dx - r dx 



r x-i _ L r dx _3 /• 

J a* + 6^ + 8 2 J x + 4 b J J 



dx 



x + 2 
(,r + 4)* 

/* 9a 2 -f 9a — 128 
J a 3 _ 5x 2 + 3^, + 9 
Put 

9^2 + 9a , _ 12 s A J? (7 

(a? _ 3) 2 (a + 1) "" (x - 3) 2 + x - 3 + x + 1 " " " ' ' ' 
whence 
9^2 + 9. T _ 128 = A (x + 1) + B {x + 1) (x - 3) + (x - 3) 2 

Put 

.a? = 3 ; then - 20 = A (4) . • . A = — 5 

x = - 1 ; then - 128 = C (16) . • . C = - 8 

a? = ; then - 128 = - 3 - SB — 72. • . B = + 17 

Substituting iu (1), 

9ar> + 9a - 128 , 5tfa? 17(to 8tfa> 



(a? — 3) 2 (a? + 1) (x — 3) 2 ' x — 3 a* + 1 

/• a 2 <7a 

'J £p4 + ^-2 _ 2 

a 2 Aa? + B C D 

+ z + zr 



x 4 + a 2 — 2 a 2 + 2 a; + 1 a? — 1 

x 2 = Ax(x 2 -1) + B (a 2 - 1) + C (a 2 + 2) (a - 1) 

+ 7) (a 2 -f 2) (a + 1) 
Put 

x = - 1; then l = C(-6).-.C=-i 

a = 1 5 then 1 = D (6) . • . D = + A 

a 2 = -2; then — 2 .= 4a?(— 3) + #(-3) 
or 

2 = 3Aa + 3B 



20 

In Trhich x is imaginary : equating tLe coefficients of th«- real 
and of the imaginary parrs, we have 

A = 
and 



B = i 



hence 



C :r dx f - | //./: /* irfj- 

j ^^— 2 "J ^^ ~ ■ ,-i -J ^n 



ilog^^i 






4. f^-^ ^iogH ** 1 ^-^ ? 1 

J y -or-4 j (*-l)(* + 2f $ 

6 - f ^ _ 7/ T r , = 2~ '^-^log(jr + 4 -271og(* + 3) 
_ r a? — Sap -f- 3 . / / - 2 

8 - JV^^TTT- *» ■ = J "°g ' - I «og I* - 2 - 1 log (* + 1 

9. / = log — ' > 

J a* + fer + B *f(a? + 2){ 

- r or" — 1 

= ^T + 15* - 6 log (jt + 1)+ 41 log i -- 

n - J Tmrzrj-] [ = ** U3i;-2(^=I) 

14 r !* = __A_ +log ^v 

J ar+2)(* + 3J 8 jr + 3 ^ B \x+2j 

16 - l — 1 — zr = — tan —7- lo § ( — r~ J 

/dx 
(*» + !) (* + *+!) 



21 

J a? 

h 



oc2 ~ 1 dx = i ioo- \ x2 ~ x ±l 

h X 2 + 1 " " } .I 2 + X + 1 

da? 



a? 2 + a 2 ) (x + &) 

1 C . / a? + & \ & . a? > 

= < logf — r ) + - tau- 1 - V 

b*+a 2 l VV(^+« 2 / a "\ 



20 



'•/. 



21 



a? (1 + # + # 2 + # 3 ) 

= log x — J log (1 + .r) — J log (1 + x 2 ) — \ tan -1 a? 

<fa^ 1 -, , . _-., 

(^-1) 2 (^ + 1) 2 " = 4(a?-l) * l0g{x i} 

+ 1 tan- 1 x - 2 \ + 1 log (.t 2 + 1) 
4 (x l + 1 ) 



oo P 2 - 1 ^ * i U 2 -aV2 + l? 

22. I aa? = log<— \ 

Ja? 4 + 1 2V2 )^ + ^Hl( 

00 /* # 2 da? 1 . ( x 2 - x V 2 4- 1 ) 

23. I = lo" 



•v/2 
a? 3 da? 



a; 2 + a? V 2 + 1 
+ <7 ^;tan- 1 (a7V2 + l) + taD '(^2-1)} 



24. J^^ = tV log (* 4 ~ * 2 + 1) - \ log (* 2 + 1) 

+ 2^3 S^o -1 (2« - V 3) - tan- 1 (2.r + V 3)j 

25 ' J ( i + x) (i + 2a?) 2 (1 + x 2 ) = 5 ' IT2^~ * l0g (1 + *° 
- t4o log (1 + * 2 ) + if log (1 + 2a?) + J tan- 1 x 



INTEGRATION BY SUBSTITUTION. 

IS. In the preceding examples, we have been able to see 
at once that, by an easily-suggested transformation, the 
given differential, or its several parts, will take forms in 
which a function of x and its differential take the place of x 
and dx in one or another of the fundamental integrals. In 
more complex examples, when this is not the case, it may 
yet happen that a function of a? may occur in the quantity in 
such a way tbat, by tbe substitution of a single letter (or 
some other function of a new variable) for that variable, an 





expression of the whole given quantity in terms of the new 
: will be of an integrable form. In this ease, we can- 
not usually foresee the new form of expression, but can only 
be guided by general views of simplification. 

13. The simplest examples are those in Mhk-h a simple 
expression in x occurs under the radical sigi e a tational 

expression in .r is another factor: thus. 



./' 



V(« + •*")•*" 



Here.it is easily seen that, it we substitute : foi < — .;:. the 

whole expression becomes a sen owers of z. 

Since 

a -f- r = z 

x = 2 — a 

= (h 

hence 

i\/ a4-x)a?dx = f % 'c - - a - 

= f- ' ] : ~ 2a f&d2 -f « 2 f r ih 

= f :- - fa : - - %i ■ z ~ 

= f (a+ a?) 1 - Ja (a + *)* + fa 2 (a + a - 
As another example, tak sre a radical occurs in 

the denominator; as, 

r .r »7.r 

J 1 + */ ;/: 
Let 

:= Vx 

x = - 
1 = 2s cfe 

'7jT f 2*22 



/f— 



-[/'•-/•*+/*.-/if-J 

= §*? - : - - _: - 21og(l + z) 

= f /- - jr + 2 y/x - 2log(l + Vx) 
Again, where nominator contains a power of x. togeth- 

er with a factor containing j.. it is useful to substitute, for x. 

~: whence z = -. This will tend to throw the power of the 



23 



variable into the numerator. For instance, 

dx 



Assume 



/ 




x 3 (x + 1) 



1 

x =- 
z 

dz 

Z 2 

dz 



r dx J z 2 r z 2 dz 

J x 3 (x + 1) "" I 1 1 + z — ~~ J 1 + z 



z 



_f. z — log (1 + z) 

11 U + 1 > 

2iX z x ( x > 

14:. A still more useful substitution than any of those 
mentioned is that of a trigonometric function of a new 
variable for the independent variable, where other functions 
of the same angle are seen to be equal to other factors (prin- 
cipally radicals) in the given differential ; but this substitu- 
tion is, in general, only useful where the whole differential can 
be expressed in trigonometric functions. Thus, in 

dx 



: 



&y[ (1 + X 2 ) 

if we put x = tan 0, the radical will become sec 0, and 

/dx _ P sec 2 e do Pcosddd 1 

x 2 V (1 -f x 2 ) ~~ J tan 2 e sec ~J sin 2 " ~~ sin 

V(l + x 2 ) 
x 
The facility with which trigonometrical transformations are 
made renders this the most successful way of dealing with 
radicals. 

15, Another substitution is that of a single letter, z, for an 
exponential function of x ; x will then be a logarithmic func- 
tion of 2, and dx will be algebraic. Thus, 

dx 



h 



1 + e z 
Let 

z = e x 

x = log z 

dz 
dx = — 

z 



24 



r dx C - r dz rdz r 

J 1 + ^ = J —-' J z(l + z)~J z ~J 1 



dz 



= log. 



1 + z 



e 



(l + <? 



16. When the method of substitution is employed for ob- 
taining the value of any integral between limits, if, for the 
limits of the old variable, we substitute corresponding values 
of the new variable, it will not be necessary to reintroduce the 
old variable in the final result, the computation being made 
from the new form of the integral ; thus, 

/dx 
x 2 4 (1 -~r 2 ] 
Put 

x = sin 6 
then 

yj (1 — x 2 ) = COS d 

dx = cos d de 

IT IT 

r 1 dx p cos do t oV - /3 

J h a?V (l-x 2 )~ J z sin 2 ^cos^ _ L _C _L ' 



EXAMPLES. 
V. 

/ " dx 
2 - f-7Tir r , = log\x + Vi* 2 -a*)\ 

J V (# — Or) 

r dx _ _ 4 (i - x 2 ) 

J 3? 4 (1 — X 2 )~ X 

4. I (3# + af x 2 dx 

1(2 i 4a 5 2a 2 n 

= 2l]l {3X+ aY ~ 6" ( + a) ' + ^" ( 

/^7^7 1 // 2 j' 2 \ 

(T+V) v (i - ^) = 72 im ~ l J\T=&) 



«)*g 



25 



xdx | 



— a 



J (a? -+• a) — 2a (a? + a)- 



* J V (a? + a) 

x 2 dx „ . 4 4a 

V 



/ar «a? * 4a , .3. _ . , j 

-^-^ = | (. + .)* - - (» + .)» + ^ (. + 4 

8. / (a + foe)*~ a? da? = ™5 f (a + foe) 2 ~ -^- (a + &a?)^ C 



9 f / 



(a? + a) 2 

(a? -fa) 2 _ . , . , „ ., . , . , « : 



— 3a (x -f a) + 3a 2 log (x + a) + 



2 v a? + a 



io. f /* =*-i /(SW'/f-V)--{ 

J ax* + bx 2 b(\l\by \l\ax 2 / x) 

H H Cx 2 V xdx _ 4 „ a _ i _ , , . 

11. J — r^ = ! a£ — S a?* + 2a; 2 — 2 tan- 1 V^ 

J 1 + x 5 3 

13. f-^L^= ? , 

J (a + foe 2 )' 2 a (a + foe 2 ) 

13 r dx - ^ 

• " J (a 2 - 



(a* — a?y a 2 V{a 2 -x 2 ) 



I v> 2 - 



14. I Via 2 — <£ 2 ) dx = 



vt a 



4 
INTEGRATION BY PARTS. 

17. Considering integration as the process of discovering 
from what function the given differential was derived, substi- 
tution is a method of discovering a function of a function 
from its differential. We now give a method of discovering 
the integral when the given differential is of a form which 
may arise from the differentiation of the product of two func- 
tions. 

Let u and v represent two functions of x ; then 
d (uv) = u dv -f- v du 
or 

uv = I udv + I v du 
Two integrals of the form 

I udv -f I v du 
will rarely occur in connection ; but, by transposition, 



j udv = uv — j v 



du 



26 

so that we can make the possibility of integrating u dv depend 
npou the possibility of integrating v du. The application of 
this formula is called itian by parts. It is useful when- 

ever, by separating the given differential into two factors (u 
and dv), we can integrate the differential factor (!. e.. find p), 
and. multiplying that by the differential of the other factor 
I du), obtain a simpler integral. Thus, given 



/ 



x cos x dx 



here, taking x tor u. and cos x dx for dv, since x is not com- 
plicated by differentiation, nor cosxdx by integration, the 
formula is of use. 

I x cos x dx = x sin x — j sin x dx 

= x sin x + cos x 
which may be verified by differentiation. 
18. Again, given 

I sin -1 x dx 

if we make 



and 

dx = dv 

then v du will be algebraic and inregrable : thus. 

I sin -1 z «a? = x sin * z — J — — 

J J V{l-a?) 

= a? sin- 1 ./- -f V(l — 
It may require several applications of this method to make 
the result directly inregrable. Thus, 



JVsi 



sin dd 

Put 

/' : - = u 
then 

v = — cos d 

du = off 2 dd 

fe 3 sin e do = — e 3 cos e + 3 / e- cos e do 



(reducing the last integral by u = £ 2 , dv = 
cos e dd) 



— _ 03 cos ^ 4_ 3 



: : sin — 2 jdsiuo de~\ 



27 

(reducing again by u = 0, dv = sin 6 dd) 

I 6 3 sin Odd = — 3 cos0 + 36 2 sine — (j\—0cos0+ j cos Odd | 

= — d 3 cos + 3 2 sin + (3 cos d — 6 sin 
19. Integration by parts is sometimes available, where the 
new integral is not more simple than the original one, but 
may be further reduced (by substitution, or further applica- 
tion of integration by parts, &c.) to a form involving the 
original integral, which can then be transposed to the first 
member. Thus, 

I cos 2 x dx = sin x cos x -\- I sin 2 x dx 

If we now continued to integrate by parts, we should produce 
the equation 

I cos 2 x dx = sin x cos x — sin x cos x -f / cos 2 x dx 

reducing to 

= 
If, however, we observe that 

sin 2 x = 1 — cos 2 x 
we have 

I cos 2 x dx = sin x cos x + I dx — I cos 2 a? dx 

which gives 

x -f- sin x cos .a; 



/ 



cos 2 x dx 

EXAMPLES. 
VI. 

,2 



C , -, x* log x x 

1. I # log x dx == — — ^ - 

J . 2 4 

2. I a? 2 sin x dx — 2x sin # + 2 coa x — x 2 cos a? 

3. / 6 sin # dd = sin — Ocosd 

4=. I x tan -1 a? dx—- - tan^ 1 # — _ 



9 9 



; ./V 



.„ , 3a? rt . sin 2x 

cos x) z dx = — — 2 sin a? -f — , — - 



28 

b. I x n log x ax = < log x — > 

_ T log (log a?) dr . . 

< . I — — = log X log (iOg X) — log X 

n C ■ . i ox — sin x cos x (2 sin 2 x + 3) 

8. I sin 4 a ^ = —^ x — ^ 

„ r . R _ 150 — sin cos (8 sin 4 <9 + 10 sin 2 + 15) 

9. / sm 6 de = * — — — — - 

J 48 

™ f ..7, 30 + sin cos 0(2 cos 2 + 3) 

10. ! co^0d0— — — — 2 — - 

r* sin -1 <r tfa? a; sin -1 .r 

11. / -- = - - + iogV(l-^ 2 

J (1 —x 2 r (1 — a?) a 

155 + sin cos (8 cos 4 + 10 cos 2 + 15) 



cos 6 6 dd = 



13. / fl! C z (IX = €"*) ^ ( 

14. fsiu- 1 x dx = x sin- 1 x + V (1 - re 2 ) 

15. / tan -1 a? dx = a; tan -1 jt — J log (1 + x 2 ) 

16. f -^~ tan- 1 * = x tan-' a; - J log (1 + a?) - t tan ~ lj? ' 2 
/ j- + 1 - J 



MISCELLANEOUS EXAMPLES. 

VII. 



V {a 2 - x 2 ) dx = x V (a 2 - x 2 ) +J ^ ^ _ 



C o- — x 2 , , , a? f a? da; 

I -i — i r d* = a sm I / , 9 — 

J V(« 2 -.r 2 ) a J V(fl 2 - 



x?) 
of) 



bv addition. 



f ^ ( tt a _ ^ fi x =, J^a- V(a 2 - /) + « 2 siu_1 ^) 

2. f V(x 2 + a 2 ) dx = i\x y/'yx* + a 2 ) + « 2 log [*+ V(^ + « 2 )]S 

3. / y ( . r 2 _ fl2) ax=i\x Vi* 2 - « 2 ) - « 2 log [x + V{* 2 - a 2 )] } 



29 

A C ^ X ^ I ^ X 

' J V(a+ bx + ex 2 ) ~~ 7c J /f ? + fo? +a A 

1 I dx 

Putting 

b 

X + 2c = Z 

this becomes, by (1) and (2) of Examples V, 

— log{2c# + & + 2yW(« + &#4- ^)i 

y C 

where the constant quantity —r- log 2c is omitted. 

y C 

5. / y* (a -\- bx -\- ex 2 ) dx 



* + 2^ = * 



this becomes, by (2) and (3), of Examples YII, 

'2ca? + b 



. ( 2c# + b , . 
•/c^ V (a + foe + or) 



4t€lC — ?> 2 

8c 2 



4- loff 



2c 



+ V (a 4- &■» + ob 2 ) > 



dx 1 2ca? — & 

O. I — T7 7— ; -T7 = -T" S1U 



VS^-f &a? — (c#) 2 j "~ -/c -v/(4ac + & 2 ) 

7. I V(a + &# — cx2 ) & x 

, { 2cx — h #/ _ ox 4ac + & 2 . . 2c# — b ) 

= V C S — i — - \/(a + &« — a» 2 ) + — — — sin- 1 -t— 7-jt\ > 

v ^ 4c v 7 . 8c 2 V(4ac+& 2 )^ 

J # V (« 2 ± a? 2 ) "~ a to a + V (a 2 ± « 2 ) 



(• + £)" 



a + fop 4- ex 2 c | ( , b V 4ac — & 2 

~ 4? 



30 

If — j-j — be negative, this becomes 

1 2cx+b-V(b 2 -±ac) 

V{b 2 - 4ac) ° g 2cx+b + V{b 2 - 4ac) 
4«c — b 2 

4c 2 " 

_j 2cx + b 



-^ — 77i — ^ e positive, then the integral is 



tan" 



V{±ac-b 2 ) V(4ac-6 2 ) 

io. p ** ^icg^^- 5 

= V(w# + # 2 ) + w* log j -/ a? + v 7 (m + x)\ 
12. J^ * 



# + V {% 2 — l) 

x 2 x^(x 2 — 1) 
= ~2~ 2 

/*2a /p 

13. / vers -1 - da? = -a 

Jo a 

14. I ar vers -1 - dx = 



+ Jlog^ + v , (^ 2 -l)i 



a 16 



15. / 3 tfx = sin -1 a — + log -/(l — ^) 



(1-a? 2 )' Vli--^ 2 ) 

# <to 5-a 3 



/* a a? 2 «# 
'Jo v 7 ^ -a?)" 16 

V{2ax - x 2 )flr=—^-- V{2ax-x 2 ) + - sin 1 ^— —J 

18. / -/(^fl.r — x 2 )xdx 

— _ j (2aa? — a? 2 )* + « y/(2ax — x 2 ) dx 

19. / v / (2a.r + ^ 2 )r7.r' 

( t r + «) \/(2ax + X 2 ) a 2 . i . „ > 

— v ^ y v v - * — - log Ix + a + j(2ax + x 2 )^ 

J-»2a 
y/{2ax — a?)a?dx 




5-a 4 



8 



21 - f-77^V^=^<* + a > + ^ to+ - a ^ 

J yf{2ax -\- x 2 ) 

22. I r == sin — tttt - 

^•J -^(l-to-a 2 ) V13 



° 1 

ol 

dx 



23 - / ^-fa + 13) = l0g ^ - 3 + ^ - to + 13) ^ 

25. / vers -1 - <?a; = na 
Jo a 

™ rV(l-i'-) . , -, sin-^fl — tff 1 log.*; 

J a? 4 OiC 3 oj? 2 3 

•* sin 2 a?tf# /a + fty _j/ V« tan # \ a? 
' J rt+Tcos 2 ^ - V «6 2 / an \ V(« + &)/ & 

28. At; 3 V(« + &^) ^ = ( " Xl'* ' ~ IsO^ + ^^ 

29 '" to 



" Jo W+ x2 ) ft 2 + z 2 ) ~ 2ab i a + & ) 

™ /*2 sin 3 x dx c 2 — 1 , „ . 2 — o 
30. / — = _^i og (i + ). + _ r 

Jo 1 + ccosa? c 3 ° ' 2c 2 



LENGTHS OF OUEVES. 

20. Let s denote the length of any arc of a plane curve, 
and x and y the rectangular co-ordinates of any point of the 

curve j then 

{dsf = {dx) 2 + [dyf 
and 

Therefore, if #i, y 1? and #2, 2fe> are the co-ordinates of any two 
points of the curve, and if 



s ors 

-1*2 Jl 

denote the length of the arc between the two points, then 
Jx 2 J *t ( \dxj S J2/2 ^'2/2 ( \dyj ) 



32T 

EXAMPLES. 

1. To find the length of any arc of a parobola measuring 
from the vertex 

y = V (4oa?) 

•i -ire?*-/ '- + *> + •» "' + y + " 

2. y = a cos- 1 f — -^\ + ^(2aa? - a 2 .) (Cycloid.) 



g 



Jo 



POLAR FORMULA. 

SI. Applying- the general equations of transformation 
from rectangular to polar co ordinates, 

a = r cos 

and 

y = r sin 

and. substituting in the formula- of the previous article, we 
have 



EXAMPLES. 

« 

1. Show that the circumference of a circle is 2xa. 

2. r = a {I + cos 0). (Cardioide.) 

Perimeter = 8a 



APEAS OP PLANE CURVES. 

22. Let x and y be the rectangular co-ordinates of any 
point of a plane curve, and x x , y u and t r 2 . i/. 2 . the co-ordinates of 
any two given points of the curve : then, if A x denote the 
area enclosed between the curve, the axis of X. aud the two 
ordinates ?/i and y 2 , 



J x 2 



ydx 



33 

and if A y denote the area enclosed between the curve, the 
axis of Y, and the two lines drawn through the given points 
parallel to the axis of X, then 



xdy 



J 2/2 

S3. In applying these forinulse, we have only to substitute 
in them the value of y or x obtained from the equation to the 
curve ; thus, the equation to the parabola is 

\f = 4ax 
whence 

y = ± 2 y 7 (ax) 

4a 

A x == ± 2 \/a I jx ax = ± —^— < a?! 55 — # 2 2 > 
Jx 2 3 ) C 



1 ^2/1 1 < ) 

rj,/^ =Wa W-**\ 



y 4a J y^ * 12a ^ x ** $ 

It may sometimes be more convenient to substitute the 
values of dx or dy obtained from the equation to the curve. 
In this case, the limits of the integral must be changed from 
a?i, x 2 , to the corresponding values of y, (y h y 2 ,) or vice versa; 
that is, the limits must always be values of the independent 
variable. 



SIGN OF THE AREA, 
24. 



J rod 
y dx will be positive or negative according as y and 
%2 

dx have the same or contrary signs. 

Now, y dx, the differential of A x , was found upon the suppo- 
sition that this area was generated by a variable ordinate, 
moviug uniformly along the axis of X, toward the right; so 
that, in this case, dx is always positive, and the sign of the 
area depends upon that of y. Therefore, if the area repre- 

y dx lies above the axis of X, it is positive ; if 

x 2 

below, it is negative, 

x dy lies to the right 

2/2 

of the axis of Y, it is positive ; and, if to the left, it is negative. 
In finding the whole area of any curve, the sum of the sev- 
eral parts is taken, irrespective of sign, or as if they were 
all positive. 
5 



34 



LIMITS OF THE INTEGRATION. 

2o. From the preceding article, it follows that, when a 

y dx changes sign : and when it 

crosses the axis of Y. / x dy changes sign : therefore, if we 

include between the limits of the former any value of x for 

which y becomes 0, or between the limits of the latter any 
value off for which x becomes 0. we shall, in general, obtain 
a difference of two areas instead of their sum. Thus, the 
equation to a carve is 

y = x* + tot 



then 



e I = I V' - 

*1 



j* Fr 



whence 



*I- 


149* 


*L- 


21* 


*.V- 


- 362§ 



and the whole area, between x — 4 and x = — S. is 
149* + 21i + 362§=533i 

If we had integrated between x = 4 and x = — 8 at one 

operation, we would have obtained 

^J_ = -192 

which is the algebraic sum of the areas. In this case, we 
might have integrated between x = 4 and x = — 4: for, 
although the ordinate at the origin is 0. the curve does 
not cross the axis of X at that point, and therefore the area 
does not change sign. Care must also be taken not to assign 
limits between which the value of the variable becomes infi- 
nite. In the following examples. A denotes the whole area 
of a closed curve. The Id always be traced. 



35 

EXAMPLES. 

VIII. 

1. y 2 — 4# + Sy - 16 

^ x J o = 26|or5J 

I 8 
-A,J o = 10§ 

2. y 2 = 4x + 8y 

]12 
= 122Jor-26£ 
o 

"I 8 
Jo 

3. 2/ = # 3 + ax<l 

A J - — 

*Jo " 12 

]° a 4 
-. = ii 

4. y 2 = 2a# — a? 2 

A = -a 2 

5. 2/ 2 = 2ax + # 2 

]2a 
= ±a 2 j3V2-£log(3 + 2V2)} 

A = f Ttab 

7. ?/ (# 2 + « 2 ) = c 2 (# — x ) 

A x J q =<?(1 -i log 2^ 

8. a 4 ?/ 2 + 6V = a 2 Z> 2 .r 2 



A = |a& 



9. Find tlie area of a loop of the curve 
a 2 ?/ 4 = x 4 (a 2 — x 2 ) 
10. Find the area of a loop of the curve 



4a 2 
T 



y 2 (a 2 + x 2 ) = x 2 (a 2 — x 2 ) — (w — 2) 

11. Find the area of the loop of the curve 

2# 2 (a 2 + # 2 ) = (a 2 — # 2 ) 2 

a 2 j3-/21og(l + V2) — 2} 



36 



12. y = x (4 — xf 



A 



]> 21 * 



13. y {a 2 + x 1 ) = a 



— n* 









A x \ 

Jo 










-•r-f 


14. 


y = sin x 




-127T 


15. 


xy — m 2 




4,]"=^ log (5) 


16. 


« 2 






y — i 

{ax - 


-X 2 ) 










^x 


ft 
n 


17. 


(y -D 2 = 


X* — 


X 2 






A *~l 


= 1 ± J { ^512 - >/27? 


18. 


(x + yf = 


:2{X 


-2/) 


19. 


y 2 — e x 










A 


J* =±2(6-1) 






A 


-]0 

J— 00 


20. 


x 2 y 2 (4:X 2 — 


■9) = 


4 

- 1 z 








# 2 





21. y — x tan a — 



A 



] 



4/i COS 2 a 
2/i sin 2a 



22. 2/ 2 (a — x) = a? 5 



= I /t sin 2a 7fc sin 2 a 



A T-± — 

*Jo 16 



37 



9q 0* ^_-, 
^iO. -5 — 77 — i 




aJ = 

Act 


:i^|V3-j: 


24.^ + ^ = 1 






J. = Tzdb 



25, 2/ 2 = & (0 - as 8 ) (tf 2 - 4) 5 



80 .„ 72 
A=-V5 + - 



POLAR FORMULAE. 



36. Let r and 6 be the polar co-ordinates of any point of a 
plane curve, and r u 6 h and r 2 , 2 , the co-ordinates of any two 
given points of the curve ; then, if A denote the area included 
between the curve and the two radii vectores y\ and r 2 , 






dd 



The same precautions (as to assigning limits) as were 
mentioned for curves referred to rectangular axes must be 
observed in the use of these formulae. Thus: required the 
whole area of the lemniscata, the equation to which is 

r 2 = a 2 cos 2 



J 02 



//2 

A =— I 



To find the whole area of the right-hand loop, we must 
integrate between the values of 



Thus, 



6 = - and 6 = — -7 
4 4 



a 2 T 4 rt , a 2 sin20-]4 a 2 



4 - l_ 4 

And as the curve is symmetrical, its entire area is a 2 . 

EXAMPLES. 

1. r = 2a cos 6 A = na? 

2. r = 2a (1 — cos0) A = 67m 2 

3. r = 2a cos cos 2d A = -77- 



38 
ABEAS OF StJBFACES OF SOLIDS OF BEYOLTJTION. 

27. If the curve revolve about the axis of X, 

Jx 2 Jx 2 ( \dx/ ) 

if about the axis of T, 

EXAMPLES. 

1. Find the surface of a cylinder. 

2. Find the surface of a cone. 

3. Find the surface of a sphere. 

VOLUMES OF SOLIDS OF KEVOLUTION. 

28. If V x denote the volume of the solid generated by the 
revolution of the area A x about the axis of X, 



- / y 2 dx 



J x 2 

and, if V y denote the volume generated by the revolution of 
A„ about the axis of T. 



= - I x 2 dy 



'2/2 



V x and V y are always positive $ so that, in assigning limits to 
the integration, it is only necessary to see that they do not 
include imaginary values of the variables. 

If we denote the volume of the solid formed by the revolu- 
tion of A x about the axis of T by V y A x , and the volume of 
the solid formed by the revolution of A y about X by V x A y , 
then 



V y A 3 



xy dx 

x 2 

Jryi 
xydy 
#2 



V X A V = 2* 

f 2/2 

29. The equation to the parabola is y 2 = 4a#; whence 



V x = 4-a, J xdx = 27tax 2 = -^x y 2 
Jo Jo * 



8- y/a x >: 4tt 



= ± -^x l y 



39 

and 

K = — - \ \fdy — -^-=- x 2 y 
Jo 16a 2 Jo 80a 2 o * 

7, A T= ± 4- Va /»* aa? = ± 

In this case, 7"^ ^i x is affected by the double sign : the positive 
sign applies to the volume formed by the revolution of that 
part of the area which lies above the axis of X, and the nega- 
tive sign to that formed by the revolution of the area below 

the axis of X. In general, I xy dx has the sign of y, and 

/ yx dy has the sign of x. It will be seen that Y y A x might 

~\y 
be found by subtracting V y from the cylinder formed by the 

Jo 

revolution of the rectangle xy about the axis of X. 



1. y 2 = 2ax — x 2 

2. a 2 y 2 -+- b 2 x 2 = a 2 b 



EXAMPLES. 




IX. 




v x 


4 

— 3 


7ra 3 


v x 


4 

— 3 


-b 2 a 


^ 


4 . 

— 3 


-a 2 b 



3. y 2 (2a — x) = x 3 

7 I T=8a 3 7r[log2-f] 

Jo 



'a 2 b 2 

V x \ = ~ab 2 

Ja O 

5. Show that the volume of a frustum of a sphere is 

^ {ft 2 + 3(r 1 » + tf)} 

in which li is the height and r L and r 2 are the radii of the ends. 

6. The curve y 2 (2a — x) = x 3 revolves round its asymptote; 
show that the volume generated is 27r 2 a 3 . 



40 

7 The carve xtf = 4ar (2a — x) revolves about its asymp- 
tote; show that the volume generated is 4z*a?. 

8. Find the volume of a frustum of a cone, having given 
its altitude and the radii of the ends. 







V= 


■• — (f^+Br 


-i? : . 


9. |^ = 


a 2 (9 - 


- x*) (x? 


— 1 : 




10. Fit 


A ~ .- ~ 




_ - • 


i. ! 


being (6 - 




Liu tne e 


srciiiul ~ — 


a). 


11. - 


— i ■ 


— , - 







: the internal radius 



j-^-*! 



CEXTEES Or GEAVITY. 

30. DeL- ::::- ", ~ :Lr ':■.:."_: r :::_ :'_r .;:> o: Y o: tiie 
ntre of gravity of the area included between any plane 
irve and the ^ : X, and by y the dist _ e of the centre 

i.'i"i;vi:v oi :Lr : : :.::. ::'_ :_r ix;s o: X; then 



, = 



■_ *-i 






and 



yd* 

, if' 



dx 



- 

J y *r 

where each integral must be obtained independent 

3 1 . For instance, it is required to determine the position 
of the centre of gravity of a semi-parabola of abscissa x and 

ordinate y, 

y 2 = 4ax 

- -2 -/a f x^dx a 

= — = - — ; = 4- X 

_ 



V« l x 

I ■ 

^1 



- 



Vjt d.r - 



41 

Here x is always positive, and y is positive or negative ac- 
cording as tbe centre of gravity of the area above or below 
the axis of X is required. 

3S. Denoting by x the distance from the axis of Y of the 
centre of gravity of a solid formed by the revolution of the 
area included between any plane curve and the axis of X 
about the axis of X ; then 

y 2 x dx 



f 

*J x< 



X 2 

x - 






>Xi 

y 2 dx 

Thus the position of the centre of gravity of a paraboloid may 
be determined as follows : 
y 2 = 4ax 

4a I x 2 dx — 

a 

3 



.. o 

4m I op dx -jy 



•s. 



And, of course, x fixes the position of the required centre of 
gravity, as the centre of gravity must lie in the axis of X. 
In the same way an integral expression may be determined 
for the distance from the axis of X of the centre of gravity 
of a solid formed by the revolution of an area about the axis 
of Y. 

EXAMPLES. 

X. 

1. Centre of gravity of a triangle. § h from apex. 

2. Centre of gravity of a cone. f h from apex. 

3. Centre of gravity of a frustum of a cone. 

h 3B 2 + 2 Br +r 2 
4 ' R? + Br + r r 

4. Centre of gravity of a hemisphere. f r from base. 

5. Show that the volume of any solid of revolution is 
equal to the area of the generating figure multiplied by the 
length of the path described by the centre of gravity of this 
figure. 

6. Centre of gravity of a semi-ellipsoid of revolution, 

prolate, f a from centre. 
oblate, f b from centre. 

7. Find the distance below the surface of the water of the 

6 



42 

centre of gravity of a quad ran tal log-ship of uniform thick- 
ness and density, with its apex just in the surface and its 
medial line vertical. .6002r 

8. Find a general integral expression for the distance of 
the centre of gravity (from the axis of Y) of the surface of a 
solid generated by the revolution of an area about the axis 
of X. 

SIMPSONS EULE. 

33, The primary object of this rule is to find an approxi- 
mate expression for the area between a curve the axis of 
X and two given ordinates (the equation to the curve not 
being known), when we can, by measurement, determine the 
lengths of the two given ordinates, and of an equidistant 
intermediate ordinate, and also the length of the com 



&■ 



'man 



interval between these three ordinates. Let 2/1 denote the 
right-hand ordinate. y Q the middle ordinate, and y' the left- 
hand ordinate 5 and let A denote the common interval. Let 
the unknown equation of the curve be 

which may be expanded by McLaarin's theorem, and written 

(1) y = A -f Bx + Cx 2 + Dx* + R 

where E denotes the sum of the remaining terms containing 
powers of x higher than the third; then denoting by A x the 
desired area, 

(2) A x = f ij dx = § (A + Bx + Cx 3 + Bx 3 + B) dx 

J -h J -h 

Of 1,3 nh 

= 2Ah + ~-^f-.+ / Bdx 
a J -h 

Xow, B is a function of x containing only powers higher 
than the third; and, when x is small, B is so much smaller 
that it may be neglected. 

B dx will consist of a series of terms of the odd powers 

-h 

of hj beginning with the fifth; and, if % be taken small, 
I B dx becomes so small that it may be neglected ; there- 
fore, assuming that h is taken small, 

A x == 2AJi + ^-^ (approximately) 

o 

(3) =*J2A+_-j 



43 

in which we have to express A and C in terms of known 
quantities. To do this, put x = h in (1), remembering that, 
as h is assumed to be small, R may be neglected ; then 

y, = A + Bh + Ch 2 + I)h 3 
putting x = — h, 

y 1 = A - Bh + C/i 2 - D7t 3 
then, by addition, 

y' + ?/ = 2A + 20/i 2 
also, putting a? = 0, y = ^4. Hence 

2C/* 2 = yi + y* - 2y 
and, substituting in (3), 

W A, = ^{%o + yi + y / } 

which is Simpson's rule. 

It is evident that, if we take two adjoining portions of a 
curve included between equidistant ordinates, and determine 
their areas by this rule, then, by addition, the whole area 

= 3 \Vi + 4 2/2 + 2^/3 + 4y 4 + Vs] 

the ordinates being marked in order from either end; there- 
fore, this rule may be extended to any odd number of ordi- 
nates; and, consequently, in determining any required area, 
by taking a sufficient number of ordinates, we can make h as 
small as we please, and our expression for the area as accu- 
rate as we please. 

The use of this rule may be extended to finding the volumes 
of solids by taking the areas of an odd number of equidistant 
cross-sections, and treating these areas exactly as the ordi- 
nates are treated in computing an area. The rule may also 
be applied to finding centres of gravity, &c, and, generally, 
to finding the approximate value of any integral. 

THE LOXODEOMIG CUBVE. 

34:t A ship sails from the equator to a given latitude, L 7 
^keeping always on the same course ; to find the difference of 
longitude between the point arrived at and the point left. 

Suppose the ship's track stereographically projected on the 
plane of the equator; then, by the principles of stereographic 
projection, the projection of the ship's track makes with the 
projection of each meridian an angle equal to that which the 



44 

ship's track makes with the meridian itself; and, also, the 
distance of the projection of the ship at any instant from the 
centre of the primitive circle is equal to B tan ^ (90 — Z), 
where L is the latitude of the ship at that instant, and B the 
radius of the sphere. 
Let d = angular difference of longitude passed through by 

ship : 
r = distance of projection of ship from the centre of 

the primitive circle ; and 
C = course. 
Then 

cot G = — ~—r- {clr being negative) 

Integrating 

9 _ r r dr 

cot G I dd = - I — 

Jo Jr r 

cot C. = log - = log cot J (90 — L)= log tan J (90 + L) 
r 

D = dif. long. = BO a= 22 tan C log tan (J L + 45°) 

but in this the logarithm is Napierian; therefore, B denoting 

the radius of the sphere in nautical miles, 

D = B tan C log £ 10 log 10 tan (J L + 45°) 



CEXTBES OF PEESSUEE. 

05. The general expression for the distance, below the 
surface of a liquid, of the centre of pressure on any immersed 
surface, is 



p = JLp_ 



where the axis of Y is vertical, and the axis of X is in the 
surface of the fluid. 

EXA3IPLES. 
XL 

1. The centre of pressure on a rectangle, with one edge in 
the surface. 

2. The centre of pressure on a trapezoid, with one of the 
parallel sides in the surface. 



45 

3. The centre of pressure of a semicircle, with its base in 
the surface. 

4. A rectangular flood-gate is 5' in height, and opens by 
turning on a horizontal axis 2' 4" from the bottom of the gate; 
how high will the water rise above the top of the gate before 
its pressure opens it ? Ans. 10' 

5. A scuttle-butt has for its base an ellipse whose diame- 
ters are 52" and 28", and for its top an ellipse whose diame- 
ters are 40" and 22", and its height is 24" $ required the 
number of wine-gallons of 231 cubic inches it will contain. 

(To be solved by integration.) 94.33 gallons. 

6. Solve example 5 by Simpson's rule, and explain why 
Simpson's rule gives an exactly correct result in this case. 

7. Find the volume of the solid cut off from a right circular 
cylinder by a plane passing through the centre of the base, 
and inclined at an angle « to the plane of the base. 

- r 3 tan a 

o 



E X A M 



IN THE 



INTEGRAL CALCULUS, 



COMPILED FOR 



THE USE OF THE CADETS 



AT THE 



UNITED STATES NAVAL ACADEMY. 



WASHINGTON: 

GOVERNMENT PKIX T I N G V F I C E 

1874. 



■iT>.-.-^>-'2> 





















^>d> 'v>-~ 






5 ^> 312] 

V5> I3> 



•;■> aw 

11 









:> ^> 



0> . ^ ^DOi>'x»Ji»'D >Z 



"7> > QPlO»> 5>3g>~> XJ* 



> ^>J> 



^ryp^ 



0> 2X> 3 









m^r-*s& 






^fe3> 



■• ^ ^^ 
■ > ■>_ 



r »2> 



3K>3T» 

Z3Bfc>>X> 

































is> -'j 2> r> ~> ■ ?:^^ -5^3g> ->3*. 2**- -^^ 

£S> J> '!> > ->^ ._?->> 'ST 

>5> » 35> s»r 

->-> t>r* J52> SDfc« "•■ 5 



33> • ? 

3B8* : 

> ^y -j > 2 



>^3x 






r^^»JaK> In? 






3»:> 



>>3 






3 >1L>- 



> 1>>; 







































22* > 3> > 

- > 3S> > 

> > 3^ •>. " 

> > 









.. 3 .^ ^ 



>■->■>> >*>>> i»^j» 



>t*Z> > .■:>" 
v* :> ^> :> 



>0> ■ 















.>: ■■■* 
> > 












:>,'>■ ^> ^>>5~>^>> 



» -■> - 

>• > 















J> ^> 



> 

u» ^> ::> > t> _- = ^'^-/ 

"D^»>T> 0>- -.153m :-^> ZZ* ; >3»- -^ "?>-> ^> ; ^ - 

^>^t> ; 3E»Ssr>^ m, ^^ ^ ^ 



-> -if > : 




~> • >> ^3> 












V^38>» 









» - 
■ j> > V 

^ » _ 



:_> _ 
^Z 

>z> 







^__ 


» ?' 






m^w& 






[ i^k> 



